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Freedericksz transition of nematics
Agnes Buka Lorenz Kramer
To cite this version:
Agnes Buka Lorenz Kramer. Linear and non-linear transient patterns in the splay Freeder- icksz transition of nematics. Journal de Physique II, EDP Sciences, 1992, 2 (3), pp.315-326.
�10.1051/jp2:1992136�. �jpa-00247634�
j phys. II France Z(1992) 315-326 MARm l99Z, PAGE 315
Classification Physics Abstracts
61.30G 47.65
linear and non-linear transient patterns in the splay heedericksz transition of nematics
Agnes Buka(~) and Lorenz Kramer(~)
(~) Central Research Institute for Physics of the Hungarian Academy of Sciences, P-O-B. 49,
H-1525 Budapest, Hungary
(~) University ofBayreuth, Institute of Physics, P-O-B. 101251, W-8580 Bayreuth, Germany (Received 2 September 1991 1991, accepted 13 December 1991)
Abstract. A three-dimensional linear analysis of transient patterns occuring during the
splay Freedericksz transition is presented including contributions relevant for the electrically driven case. The most rapidly growing modes have a wave vector predominantly along the initial director alignment. The onset of the pattern formation is compared with the results of a
nonlinear calculation carried out previously for the
wave vector in the perpendicular direction.
This shows that the nonlinear effect is the relevant mechanism for large dielectric anisotropy whereas for low anisotropy the linear pattern is favored. Preliminary experiments have been carried out with electric and magnetic fields using two materials with high and low didectric anisotropy. Both, linear and nonlinear structures were found. A transition between the two types of patterns induced by changing the frequency of the applied electric field is presented.
1. Introduction.
The occurrence of transient patterns appears to be a general feature of low molecular weight
nematics when a Freedericksz transition is induced by a sufficiently high magnetic or electric field jump. Two different types of transient patterns have been observed so far during the splay
Freedericksz transition iri a planarly aligned nematic slab. One of these structures was first detected in the magnetically driven case. It is a pattern composed of somewhat irregular stripes
oriented predominantly perpendicular [1, 2] or obliquely [3, 4] to the initial planar director
alignment. This phenomenon can be explained on the basis of the nematodynamic equations by the fact that due to backflow the linear destabilising modes with nonzero wavenumber
(perpendicular or oblique depending
on the conditions) are the most rapidly growing ones [1, 3].
Although in the electrically driven case the existence of transient domains has been known for some time [5], a pattern consisting of Stripes parallel to the initial director alignment was
identified only recently in the material SCB [6, 7]. This second type of transient structure
cannot be explained on the basis of a linear theory. However, a theory where the pattern- ing process was treated superimposed on the homogeneous Freedericksz realignment so that
nonlinear coupling terms arise provided a qualitative explanation of the phenomenon [8]. The decisive mechanism for this pattern formation is connected with the field distortion due to the strong anisotropies. A sufficiently high ~alue of the anisotropy can easily be realized for the dielectric permittivity and/or conductivity. The diamagnetic anisotropy on the other hand is usually too low and consequently the magnetic field remains practically undistorted iri the
elastically deformed nematic. Thus in a magnetic field one indeed expects only transient pat-
terns of the first type with a wave vector along the initial director alignment contrary to the
electrically driven case where one should be able to induce both types of structures by choosing compounds with small or large electric anisotropy or by changing its value with an external
parameter like the frequency.
Experiments where both types of patterns with a cros8over between them are seen on the
same sample are yet missing. A18o the theory is not yet in a stage to describe the transition since the linear analysis has not been done for the electric case where the field distortion and
conductivity effects have to be included. Moreover the calculations have to be extended into three dimensions. Here we will attempt to provide some of these missing links.
2. Theory.
We consider a planarly aligned nematic slab of thickness d. The plane of the slab is chosen to be parallel to the z y plane with the director aligned along z. We present a 3-dimensional
linear analysis of growth rates of modes with
an arbitrary wave vector (q,p) in the z y plane.
Our starting point is the full set of linearized nematodynamic equations as given in equations (3.2amf) of reference [9], neglecting flexoelectric effects and the time derivatives acting on the
velocities, I-e- fast modes. Thus one has a system of six coupled differential equations for the electric potential 4l of the field distortion, for the two components of the director fly and nz (or equivalently two angles) and for the three velocity components v~, vy and vz. These equations
describe the behaviour of a nematic in an applied electric field as a function of the frequency.
We use this complete form to evaluate the growth rates.
Applying the lowest-order frequency expansion, where it is assumed that the director cannot follow the oscillations of the a-c- voltage, the time dependence of the variables (electric pc- tential, components of n and v) after switching on the external field is essentially exponential
with a growth rate a -The magnetically driven transition can then be interpreted as a special
case I-e- one uses the final results obtained for the high frequency (or conductionless) elec- tric field and one has to replace the dielectric anisotropy by the diamagnetic one. We treat the z dependence together with the realistic rigid boundary conditions by applying the test function approximation as used in [9, 10]. Then the electric potential and all components of the velocity can be eliminated, which leaves us with two coupled algebraic equations for nz and fly. For simplicity we set here the Leslie coefficient a3
" 0, which is a good approxima-
tion for our type of problem except for (q,p) - 0, see [il. We measure a in units of I/rd
where rd
=
~id~/(Kiix~) is the director relaxation time, wave numbers in units of xId and in- troduce e
= V~ /(( I where l~h " x Kii/(coca) is the Freedericksz threshold. Then we get:
fl~Bi~
+ Pi~~)~
'j
pnz + fl~(B~~
+ «) + Pi j~p~ ~«j
fly = ° (1)
3
N°3 LWEARANDNON.LINEARTRANSIENTPATTERNS 317
fl4~12P~ fl3~~j~ ~ + '~9g~ + (~ ~ )~9~g~j 'lz + fl4(~22
+ ') fl3 )~ PflV
~ (~)
3
where
Q = Cssle[ (3)
c = ejq2 + Q, Q = i + q~ + p~ (4)
~S ~q2 + ~qj2 + ~2~2c2 ~~~
A = al + e[rod/rd, B = I + rod/rd (6)
'1" 'al'li ~l " ~a/~11 Kli
" K,, /Kll ~'ld To
# gag i la1 (7)
The B;k-s are energy densities and the fl;-s are viscous force terms given in the Appendix.
The integrals I; are also listed there. Note that Q
= 0 for row » I where To is the charge
relaxation time. The solvability condition of equations (I) and (2) gives the growth rate
a(q,p). The procedure is analogous to that presented in reference [9, 10] for the calculation of the neutral curve which is obtained from the condition a
= 0. In the usual case when
a < rd/ro equations (6) reduce to A
= a[, B
= I and one obtains a quadratic equation for a which can be evaluated for any q, p and e.
We consider now the simple case of p = 0, I-e- wave vector parallel to the director, where equations (I) and (2) become independent of each other. One obtaines a by setting the
coefficient of nz in equation (2) equal to zero. Under the condition
a « rd/ro this gives :
«(q) =
e 1<~q2 ssq~~~ ~ ~~ ~ ~~ ~
~~~?/fl3
~~ ~~ ~ ~~~~~
~~~
where now
~ " ~~ + ~~~
(i + q2 +
«iqlil~i@~+
q2 + eiq2)2 ~~t~~ ~~~
Expression (8) is analogous with equation (8) in reference [I] except that we considered electric field instead of magnetic and used approximate rigid boundary conditions instead of
free ones. Moreover we have from the beginning neglected a31a2. Our expression contains
two additional terms in the numerator proportional to Ss and Q, respectively. The first one describes the effect of the field distortion due to the dielectric and /or conductive anisotropies which is neglegible in the magnetic case. The other term incorporates the flow which is induced
by electrohydrodynamics, I-e- space charge effects. This term vanishes for zero conductivity,
for sufficiently high frequency, or if the electric field is replaced by a magnetic one.
One can see that in the stated approximation (a31a~ = 0) the homogeneous deformation (q = 0) always corresponds to a local maximum of a with height e (we remind that we consider
the situation El > 0). However the backflow term proportional to q~ (for small q) in the
denominator has the effect that above some value of e a second maximum of a develops at q = qmax # 0 which becomes larger than a(0) at a higher value em. This is usually considered
to be the criterion for the onset of pattern formation in the z direction and determines qmax under the assumption that the maximum of a as a function of p lies at pmax = 0.
It is easily seen that the new term proportional to Ss in equation (8) suppresses the tendency
for pattern formation. Thus em increases strongly with increasing e[ and al and even diverges
when the anisotropy becomes sufficiently large.
The term proportional to Q on the other hand enhances the tendency towards pattern for- mation if (a[ El) > 0 and can for sufficiently low frequencies even lead to ecz < 0. This then indicates that one has a situation of permanent pattern formation due to the electrohy- drodynamic effects. In such a situation the pattern may vanish again for larger e. Permanent
patterns can arise also outside this regime (I.e. for em > 0) at sufficiently high fields, as an instability of the Freedericksz distorted state (simiar to the case of the bend geometry [11]), but the linear theory cannot deal with this situation.
In figure I the dependence of em on the relative dielectric an18otropy El is shown for fre-
quencies such that row » I. The other material parameters used are essentially those of SCB which are actually very similar to those of MBBA. They are listed in the figure caption. em increases rapidly with El starting from a finite value at El = 0 which is the relevant quantity for the magnetically driven case, see [I], and diverging at El = 2.8 (near the divergence one has em = 1060/(2.8 e[)~ and q$~
= 26.5 /(2.8 El)). For very large values of em the assumption
a « rd/ro actually breaks down, but this region is of no interest. A18o shown in Fig-I is the estimate of the onset of the transient pattern ecy iri the y direction (q = 0) calculated
previously, see [8], as a result of nonlinear effects.
In figure 2 the same quantities are plotted for the low frequency linfit row < I. In this case
al becomes relevant and we show the plot for two values. Again ecz increases with increasing El
now starting from em = -I at El = 0 (see insert). The fact that em - -I for El - 0 merely
means that the threshold for the permanent electrohydrodynamic instability remains finite
while l~h (the Freedericksz threshold) diverges. The small range of El where em is negative
describes the region of permanent electrohydrodynamic convection. This range increases with
increasing a[. The most interesting feature is that em and ecy cross each other as a function of
El with the crossover value of El depending on frequency. Thus for a given material one may
be able to induce any of the two different structures by changing the frequency.
We should point out that, (except for very small e[ and with appropriate values of the other material parameters) the most rapidly growing linear mode for qmax # 0 does not occur for p = 0, but one usually has pmax # 0 (see a18o Ref. [3]). For small El the ratio ofpmax/qmax is
rather small (it is 0.23 at El = 0.04) and increases with El. The difference between ec at p = 0 and at pmax # 0 is always very small, so we do not wish to discuss this complication although
all the information can be extracted from equations (I) and (2).
For completeness we also make contact with the case q
= 0,p # 0. Then the solvability
condition of equations(I) and (2) reduces to
(~ill +')(~22 + ') ~~2P~ " ° (~°)
which leads to
a = -b + @~, (it)
b = )(Bii + 822) = ((-e + ISK[~) + j(I + K[~)p~ (12)
C " ~ll~22 ~~2P~ " (~~ ~ ~~2P~)(~5~~2 ~P~) ~((~ ~~Z)~P~ (~~)
This result is identical to the lowest order (linear) relation found in [8], equation (19). It shows that for materials with K[~ > 12/(12+ /@ m 0.30 which is the case of usual nematic8
N°3 LINEARANDNON-LINEARTRANSIENTPATTERNS 319
40 high frequency
s~ ~cx
30
20
e~
lo
0
0 2 3 4 5
E~,
Fig-I. The critical control parameter ec denoting the onset of the pattern formation in both
x and y directions
vs. the relative dielectric anisotropy El, for &equencies such that row > 1.
Kll " 6.5 X10~~~N, K22
" 4 X10~~~N7 K33
" 9 X I@~~~N, al
" a3 " 0, 1Y2
" ~90 Cp, 1Y4 =
90 cp, a5 = 65 cp, a6 = -35 cp.
ZOO
lo11J
~c frequency
~°
,,
~
~~~ /
~0 '~,l
O-Z
i ~U(=0.17
aj = o.50
0
0 1 2
,
3 4
E~
Fig.2. The same as figure for frequencies such that row < 1 for two values of the relative conductivity anisotropy «(.
and substances considered here a decreases with increasing p~ and patterns (with q = 0) do not appear on the linear level. For materials with K[~ $ 0.30 the pattern would be permanent [12].
The growth of transient patterns with q = 0 can only be described with nonlinear effects,
I-e- in conjunction with a homogeneous reorientation of the director. In principle one should
take into account this nonlinear effect also in the other directions, but the calculation seems to
be a difficult analytical task. We are planning such an extended analysis in order to provide
a description of the dynamics over the whole lifetime of the transient pattern including its final decay. Experimentally it is probably difficult to distinguish the two mechanisms. A
signature is, however, the initial exponential growth of the linear process in contrast to the
more complicated time behaviour of the nonlinear one, see [8]. Thus a detailed measurement of the time behaviour of the contrast and that of an integral quantity like the capacity should
allow a distinction.
During the growth of linear modes with (q,p) # 0 there are adjacent regions in the sample with opposite director orientation. Since the system ha8 an z - -z symmetry this antisym-
metric distortion
can not decay spontaneously into the homogeneous Freedericks2 distortion where the symmetry is broken on a large scale. One might therefore argue that the structure evolves into an array of domain walls which eventually annihilate in a slow process. How-
ever this picture is probably not correct since the z - -z symmetry will be broken from the beginning on by the homogeneous mode which is also growing, competing and eventually winning.
The criterion for the onset of pattern formation has to be treated with some caution because the experimental imageing process of the patterns involves spatial variations of the refractive index of the medium and therefore puts a bias on large wavenumbers. In fact the homogeneous
reorientation corresponding to 2ero wavenumber does not lead to any contrast in the experi-
ment. Work on including properly the imageing process is in progress.
3. Experiments.
Two room temperature nematics, 4-pentyl -4'-cyanobiphenyl (SCB) and 4,4'-dibutyl- asoxybensene (dBAB) with similar thermal, elastic, viscous and diamagnetic properties but with different electric parameters have been used in the experiments. The difference in the relative dielectric anisotropies of the two substances is about two orders of magnitude, El = 2 and 0.04 for SCB and dBAB, respectively. The relative conductivity anisotropies are a[ = 0.5 and 0.17, respectively. These differences in the dielectric and conductivity anisotropies enabled
us to experimentally demonstrate transient patterns with wave vectors parallel to z or y as
well as a crossover between them.
Sandwich cells of size 25 mm x 25 mm made out of conductively coated glass plates have been prepared with a spacing of100 pm. Planar alignment (in the direction of z) of the nematic director was achieved by rubbing a polyimid coating. Experiments were carried out at a stabilized temperature of 22° C. The splay Freedericksz transition was separately induced
with static magnetic and harmonic electric fields in the frequency range of 5 20000 Hz.
The image of the patterns induced by an electric field has directly been observed and recorded in the central part of the cell (approximate area is I mm~) in a polarizing microscope with the
polarizer adjusted parallel to the initial director alignment. A laser diffraction has also been
performed on the transient structures in order to get information in the case of the magnetically
driven transition where no direct microscopic observation of the patterns was possible.
Transient patterns were recorded in SCB during the splay Freedericksz transition induced
by a harmonic electric field. They consist of a set of stripes with a preferred direction, and appear always above a critical field (corresponding to ec) larger than the Freedericksz threshold.
Detailed experimental studies (time and voltage dependence of the contrast of the pattern, etc.)
were given elsewhere [7]. No qualitative change was detected as a function of the frequency.
The preferred direction of the lines appears to be parallel to the initial director alignment z,
which means qmax
= 0 and pmax # 0. A typical pattern taken at the moment when the contrast reaches its highest value is shown in figure 3. Transient patterns are always somewhat irregular
Tone compares them with permanent structures for example Williams domains. The reason
is probably that the lifetime of the pattern is too short for a proper wave number selection.
Note also the high value of the control parameter one has to use in order to get a reasonable contrast (e m 24 for Fig. 3) compared with EHD patterns where one usually has e m I.
The image of the pattern has been digitized with 256 grey scales and a resolution of 512 x 512 pixels. The array of the intensity data has been processed: id Fourier transforms (FFT)
N°3 LINEARANDNON-LINEARTRANSIENTPATTERNS 321
x
~y
Fig.3. A transient pattern taken
on SCB in electric field, for e
= 24 and f = 65 Hz. The picture corresponds to an area of 0.5 mm x 0.7 mm in real space.
p
a.
(
» -
c
o to m m m «
~
~ Z
b. ~ g
» -
c
o Jo m m «
Wavenumber
FigA. The Fourier transforms of the pattern shown in figure 3: (a) along x; (b) along y. The wavenumber units are: (a) 2K mm~~ (b) 2K/0.7 mm~~
were performed along lines in z and y direction, respectively, and averaged over the total number of lines. Figures 4a and b show the Fourier transform of the intensity functions along
z and y respectively. In the z direction the spectrum is structureless, its amplitude decreases monotonously. In the perpendicular direction the Fourier transform shows a distribution with
a broad maximum at
a finite value (m 10) which corresponds to the preferred wavelength
m 70 pm. (The peak structure results from the insufficient sampling and has presumably no significance. The sharp maximum at zero wavenumber has also no physical importance, it is the result of the nonzero intensity average.) Similar, but less detailed information is contained