Pattern formation and instability of smectic-A filaments grown from an isotropic phase
Hiroyoshi Naito,
1
Masahiro Okuda,
1
and Ou-Yang Zhong-can
2
1
Department of Physics and Electronics, Osaka Prefecture University, Sakai, Osaka 593, Japan
2
Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China
Received 16 October 1996
Pattern formation and instability of smectic-A filaments grown from an isotropic phase have been investi-
gated. The evolution and shape equations for the filaments are derived by a variation of the free energy of the
filaments. It is shown from the evolution equation that the length of the filament increases exponentially with
time under a cooling process. It is also shown from the shape equation that the existence of the threshold length
of the straight filaments for buckling and the global properties of the filaments can be predicted. The theoretical
results are demonstrated experimentally in the binary mixture of octyloxycyanobiphenyl with dodecyl alcohol.
S1063-651X9715102-9
PACS numbers: 61.30.Cz, 02.40.Hw, 46.30.-i, 82.65.-i
Smectic-A liquid crystals Sm-A LC’s appear in a variety
of shapes on cooling the high temperature isotropic ( I ) liquid
phase 1. The shapes have a very complex topology, and
exhibit a great deal of thermotemporal evolution. Such self-
organizing processes in the Sm-A phase are intriguing, and
therefore have attracted a number of studies 2–8. The most
interesting example of the processes is the formation of Sm-
A filament structures and their thermotemporal evolution
2,5,7. The Sm-A phase first appears in the form of fila-
ments in various Sm-A LC’s as the I liquid is cooled below
the I –Sm-A transition temperature 2–5,7,8. The filaments
grow rapidly in length but not in diameter. During growth,
the filaments continuously buckle to take on a serpentine
form, and subsequently become extremely convoluted and
nearly space filling. The filaments are metastable, and even-
tually transform to compact domains 7.
Most of the general features of the growth of Sm-A nuclei
are much different from those of the solidification of con-
densed matter such as snowflakes and ice crystals 9. Thus
studies of the formation of Sm-A filament structures and
their thermotemporal evolution are very important for under-
standing pattern formation and instability in soft matter or
complex fluid.
We have developed a theory for equilibrium shapes of
Sm-A nuclei grown from the I phase 6,7. Within the frame-
work of our theory, a filamentary structure is regarded as a
Sm-A tube with thickness of
o
-
i
, where
o
and
i
are the
outer and inner radii of the tube, respectively. Since the Sm-
A phase exhibits a layer structure in which the average mo-
lecular orientation is parallel to the layer normal, the fila-
ments consist of concentric cylindrical Sm-A layers 2,5,6.
In this paper, we propose dynamical and static theories for a
description of the dynamics and shapes of Sm-A filaments
grown from the I phase. The evolution and shape equations
are derived from a variation of the free energy of the fila-
ments under cooling processes, and the growth behavior of
the filaments is predicted. The predicted results are compared
with the experimental results in the binary mixture of octy-
loxycyanobiphenyl 8OCB with dodecyl alcohol DODA.
The LC material used here was the binary mixture of
8OCB and DODA, whose phase diagram was reported by
Pratibha and Madhusudana 5. In this system, the nematic
phase is suppressed for a molar concentration ( 20%) of
DODA, and the I and Sm-A phases coexist in a fairly wide
temperature range for example, 42–30°C for 60% of
DODA studied here. The LC cells of dimensions 1010
mm
2
and of thickness 50 m bounded by glass plates were
prepared. The surface of the glass plates was not treated the
observed growth pattern was not affected by the rubbing of
the plates. The sample temperature was controlled in a hot
stage Mettler FP82HT with a processor Mettler FP90.
The growth processes and the equilibrium shapes were ob-
served with a polarizing microscope Nikon equipped with a
charge-coupled-device video camera Sony DXC-151A.
The images were recorded and fed into an image processor
Shimadzu Nexus600.
The cells with 40% of 8OCB were cooled from the I
phase at -0.1 °C/min, and the cooling was stopped at
36.4 °C in the coexisting region for the observation of equi-
librium Sm-A shapes. Figure 1 shows the initial stage of the
growth sequence of a Sm-A filament from I phase in the
mixture of 8OCB and DODA at a cooling rate of -0.1 °C/
min. As stated above, the filament finally collapses forming
compact domains at the LC/glass interfaces 7. We have
reported that the compact domains have toroidal structures
with a ratio of generating radii of 1/ 2 1 7, but in rare
cases a circular toroidal filament whose ring width is al-
most equal to the diameter of the straight filaments is found.
Figure 2 shows an equilibrium circular Sm-A filament grown
from an I phase observed at 36.4 °C.
A Sm-A filament of length l can be described by a tube
with the outer and inner surfaces,
r
o
=r
s +
o
n
cos +b
sin ,
1
r
i
=r
s +
i
n
cos +b
sin ,
respectively, where 0 s l ,0 2 , r
is the axial curve
of the filament, and n
( s ) and b
( s ) are the normal and binor-
mal vectors of r
, respectively 10from the experimental
results in Fig. 1, it is reasonable to assume that the radii
o
PHYSICAL REVIEW E FEBRUARY 1997 VOLUME 55, NUMBER 2
55 1063-651X/97/552/16555/$10.00 1655 © 1997 The American Physical Society