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Morphological phase transitions in viscous fingering patterns in the liquid crystal 8CB
A. Buka, P. Palffy-Muhoray
To cite this version:
A. Buka, P. Palffy-Muhoray. Morphological phase transitions in viscous fingering patterns in the liquid crystal 8CB. Journal de Physique, 1988, 49 (8), pp.1319-1323. �10.1051/jphys:019880049080131900�.
�jpa-00210813�
1319
Short communication
Morphological phase transitions in viscous fingering patterns in
the liquid crystal 8CB
A. Buka(*) and P. Palffy-Muhoray(**)
Department of Physics, University of British Columbia, Vancouver, B.C. Canada
(Reçu Ie 27 aoit 198?’, rivisi Ie 20 mai 1988, accepti le B4 mai 1988)
Résumé. 2014 Nous presentons un diagramme de phases pression-température pour la morphologie des figures de digitation visqueuse observées dans les phases isotrope, nématique et smectique A du ? liquide 8CB. En plus de la structure branchée dense, nous observons deux régimes dendritique,
distincts dans les phases nématique et smectique. Nous avons étudié la largeur caractéristique des doigts en fonction de la pression et nous avons également considéré les effets d’alignement par une surface ou par un champ magnétique.
Abstract.2014 We present the morphological pressure-temperature phase diagram for viscous fingering patterns observed in the isotropic, nematic and smectic A phases of the liquid crystal 8CB. In addition to the dense branching structure, two distinct dendritic regimes were observed in the nematic and smectic phases. The dependence of characteristic finger width on pressure was studied, and the effects of surface and magnetic field alignment were considered.
LE JOURNAL DE PHYSIQUE
J. Phys. France 49 (1988) 1319-1323 AOUT 1988,
Classification
Physics Abstracts
05.70L - 61.30 - 68.70
Non-equilibrium pattern formation is of considerable current interest, due to the simi- larity of underlying mathematical structure in a
wide variety of different systems [1]. A question
of central importance is the role of anisotropy
in pattern selection [2]. The role of anisotropy
has been investigated both in computer simu- lations [3] and in viscous fingering experiments using isotropic fluids in anisotropic Hele-Shaw
cells [4, 5]. More recently, because of their in- herent anisotropy, liquid crystals were used to study the pattern morphology of viscous fingers
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049080131900
1320
[6-8]. In this paper we present the first complete morphological pressure-temperature phase dia-
gram for an anisotropic fluid.
We have studied pattern formation in a
thermostatted radial Hele-Shaw cell containing
the liquid crystal 8CB (4,4’-n-octylcyanobiphe- nyl) into which air was injected at constant pres- sure. Details of the experimental arrangement
are given in reference [9]. We restrict ourselves to the viscous flow regime, that is, to a region of
P- T space where inertial effects are negligible.
The cell temperature was regulated to within
z 0.2° C in the range of 20-50° C, and the ap-
plied pressure was varied from 0-60 kPa with an
accuracy of ± 1 kPa. The growth of the pat-
terns at constant temperature and pressure was recorded on video-tape.
The observed patterns fell into three cate-
gories :
(a) dense branching (DB) : a dense, space-filling pattern characterized by unstable tips which bifurcate repeatedly during pattern growth;
(b) sparse dendritic (SD) : a less dense pat-
tern characterized by a fixed number of straight fingers with stable tips and short side-branches at relatively large angles (about 60° ) to the di-
rection of the fingers; and
(c) dense dendritic (DD) : a pattern with density greater than that of SD, characterized
by a fixed number of main branches with relati-
vely stable tips and narrow side-branches with
a wider range of lengths and angles than SD.
Fully developed patterns typifying these mor- phologies are shown in figures la-c.
The pressure-temperature phase diagram
for viscous fingering patterns in 8CB is shown in figure 2. The morphological phases were de-
termined by visual examination of the pattern growth recorded on video-tape. We have also
carried out digital image processing and analysis
of these patterns [10].
In the isotropic phase, above TNI = 40.5° C,
we see only the DB morphology. Above the pres-
sure of 25 kPa in the isotropic and nematic
phases inertial effects start to become impor- tant, and the system is no longer in the viscous
fingering regime. The stability of the DB mor- phology in the isotropic phase is discussed in references [7, 9].
Fig. 1.- Typical pattern morphologies : (a)
dense branching (DB), (b) sparse dendritic (SD), and (c) dense dendritic (DD).
Fig. 2.- Morphological pressure-temperature phase diagram of viscous fingering patterns in 8CB.
In the nematic phase, the director align-
ment in the cell is initially homeotropic. As pres-
sure is applied, flow alignment causes the direc-
tor field to become radial [9]. Immediately be-
low TNI, the morphology is DB, and the pattern is indistinguishable from that in the isotropic phase. As the temperature is further lowered,
stable dendritic tips appear and signal the on-
set of the SD morphology in the region of P- T
space indicated in figure 2. At both high and
low pressures, the tips of the dendrites begin to
split, and the SD regime disappears. We specu- late that this is caused by lack of flow-alignment,
as well as by inertial effects at high pressures.
The temperature dependence of the pat-
tern morphology in the nematic phase has been
discussed in reference [9]. Here we have attri- buted the disappearance of the SD regime as
the temperature is lowered to the destruction of flow alignment by smectic fluctuations. It fol- lows that, in a material which does not exhibit a
smectic phase, the SD regime should persist as
the temperature is lowered until crystallization.
To verify this, we have studied viscous fingering patterns in the liquid crystal 7CB, which does
not show a smectic phase. As expected, we have
found sparse dendritic patterns persisting on co- oling until solidification.
In the smectic A phase, at low pressures the DB morphology and at higher pressures the DD
morphology are observed. The DD regime ap-
peared immediately below the nematic-smectic transition. Due to the significant increase of vis-
cosity in the smectic phase, pressures well in ex- cess of 60 kPa could be applied without evidence of intertial effects. This DD phase has not been
observed in previous investigations [8].
The above morphological transitions are
not abrupt, and there exist regions in P- T space
where a single morphological phase cannot be clearly identified. These regions of intermediate
morphology have been left blank on the phase diagram.
We have attempted to estimate a charac-
teristic finger width in the DB regime. In or-
der to do this, we have constructed arcs (cen-
tered on the point of air injection) tangent to each inlet separating the fingers and extending
from one inlet to another. Such arcs on a sec-
tion of a typical DB pattern are illustrated in
figure 3. We found that for a given pattern, the lengths of these arcs do not depend on the ra- dius, and that their standard deviation is typi- cally less than 20% . We assume that the average
arc length gives the scale of the tip-splitting in- stability, and we define a critical finger width Wc equal to one half of this quantity. We found
that at low pressures Wc is essentially indepen-
dent of temperature in the nematic and isotro-
pic phases, however, it decreases strongly with temperature in the smectic phase. At constant temperature Wc decreases with pressure in the isotropic and nematic phases, but remains
nearly constant in the smectic phase.
Fig. 3.- Outline of a section of a typical DB pattern. The dashed lines indicate arcs used in the calculation of critical finger widths.
It is possible to make a connection between the pressure dependence of Wc and the results of linear stability analysis. If the kinetic term in the boundary conditions is ignored, the wave
number m* of the fastest growing mode is given by [7]
where Po is the applied pressure, o, is the surface tension (a _ 10-2 Pa my is the radius of the inner liquid-crystal - air interface and Ro
is the cell radius. There are two distinct ways of viewing the stability of fingers in a well- developed pattern.
If each finger-tip is considered separately as
a semi-circular interface unaffected by the proxi- mity of its neighbors, then at the onset of the instability when the tip splits, R = Wc , m* = 4 and, from equation (1)
Alternately, as hypothesized by Ben-Jacob et al.
[4], one may view the outer perimeter of the pattern as a circle with radius Rp on which (large) perturbations exist. Equation (1) then gives the wave-number of the fastest growing
mode among these, assuming that linear stabi- lity analysis holds. In this case
1322
In order that the pressure field have the correct value at the interface, the surface tension in
equation (1) must be replaced by its effective value orpl Wc ; this gives
Euperimental results indicate that Po is proportional to We-3 as shown in figure 4. The slope of the best fit line is 4.13 Pa, while equa- tion (3) gives 4.14 Pa, in close agreement.
Fig. 4.- Graph illustrating the relation between
applied pressure P, and critical finger width Wc.
T = 41.5° C. The solid line is the best fit.
In general, in the isotropic case, the cha- racteristic finger width Wc is independent of
the viscosity and, for dimensional reasons, can
only depend on the ratio Polo, . Experimentally,
we have found that Wc decreases with pressure for all temperatures above TNI. Since the sur-
face tension usually decreases with temperature,
we would expect the finger width also to de-
crease with temperature. We see no evidence of this in our system; however, the surface tension
changes only slightly in the isotropic tempera-
ture range covered. (We have used the capil- lary method to measure the surface tension of
8CB, and have found that it is constant to wi- thins 10% within the entire temperature range
studied.)
In the nematic and smectic phases, Wc still decreases with pressure, but this dependence be-
comes weaker as the temperature is lowered. Ho- wever, Wc decreases dramatically with tempera- ture ; it is at least one order of magnitude smaller
in the smectic phase than in the isotropic. This
behavior is opposite to what would be expected
from the temperature dependence of the surface tension and can only arise through the tempe-
rature dependence of some dimensionless quan-
tity. Such quantities may be formed from princi- pal values of the effective viscosity tensor or the
surface tension tensor. Since it is well known [11]
that the viscosity coefficients. show very strong temperature dependence as the smectic phase is approached from above, we expect the viscosity
to play the dominant role in determining stabi- lity and in finger width selection.
In addition to homeotropic alignment, we
have also investigated homogeneous alignment
where the director is parallel to the plates. This alignment was affected by both surface treat- ment of the glass with PVA and buffing, and by
the application of a 2.5 kG magnetic field. We
have found that this initial alignment had no
effect on the pattern morphology in the nema-
tic phase, as the flow immediately realigned the
director field radially. In the smectic phase, ho-
wever, the initial alignment persists with consi- derable effect on the resulting pattern as shown in figures 5a-e. At small pressures (and low velo- cities) the flow is essentially parallel to the smec-
tic layers. In this direction, we assume the fluid
to show Newtonian behavior. In the direction
perpendicular to the layers (along the direction of the initial director alignment) we expect the viscosity to be strongly shear dependent due to
the permeation flow [12] associated with inter-
layer hopping of the constituent molecules. We expect therefore to see a nonlinear dependence
of average velocity on pressure gradient. This ef-
fect can be seen from the direction of the long
axis of the pattern envelope which changes with
pressure from perpendicular to parallel to the initial alignment direction as shown in figures
5a-d. This effect has been observed previously
[8]. It is interesting to note that the onset of the tip-splitting instability is different in the two di-
rections ; the interface is unstable in the direc- tion perpendicular to the initial alignment even
at very low pressures, while the instability in the parallel direction does not occur until the pres-
sure reaches about 10 kPa. At higher pressures,
only the tips of fingers which grow in the per-
pendicular direction become stable. As expected
from the flow geometry, this occurs in the same region of P- T space where the DD morphology
exists in the case of homeotropic alignment.
Fig. 5.- Viscous fingering patterns in the homogeneously aligned smectic phase. The
( --->) symbol indicates the alignment direc-
tion. T = 23.0°C. (a) P = 3.0 kPa, (b)
P = 6.0 kPa, (c) P = 15.0 kPa, (d) 20.0 kPa,
and (e) 30.0 kPa.
In conclusion, we have determined the P- T
phase diagram for viscous fingering patterns in the liquid crystal 8CB. We have found that den- dritic regimes characterized by stable tip growth
occur at low temperatures, where the bulk pro-
perties of the liquid crystal are strongly aniso- tropic. The pressure-anisotropy phase diagram
of Ben-Jacob et al. [4] for an isotropic fluid with
an anisotropic grid engraved on one of the plates
is in qualitative agreement with our results near
one portion of the SD regime; increasing their anisotropy parameter corresponds to reducing
the temperature in our case. (We have not inclu-
ded the regions of stable circular pattern growth
on our phase diagrams at extremely low pres- sures, which correspond to the faceted growth
of Ref. [4].) We have determined that, in 7CB,
which does not exhibit a smectic phase, the SD regime is stable, as expected, until solidifica- tion. In the isotropic phase of 8CB, the charac- teristic
finfer
width is found to be proportionalto (P / u) - , in agreement with the predictions
of linear stability analysis. We argue that, for di-
mensional reasons, anisotropy must be respon- sible for the strong temperature dependence of
the finger width at lower temperatures. Finally,
we have made observations regarding the growth
and stability of fingers in the aligned smectic phase.
Acknowledgment.
We are grateful to the Natural Science
and Engineering Research Council of Canada for financial support.
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