7. Out-of-plane tumbling-wagging composite state with periodic chirality
(OTP): In this nonplanar mode, the bulk director dynamics is planar and
rotational, and in the two boundary layers, it is nonplanar and rotational, and
in the two boundary layers, it is nonplanar oscillatory. The spatial profiles of
the periodic director motion are anti-symmetric. The director field exhibits
periodic chirality, that is, after a cycle of 2π rotation of the bulk director, the
system periodically recovers the spatially homogeneous director configurations (i.e., n ffi (1, 0, 0)) for 0 y
Ã
1
8. Out-of plane tumbling-wagging composite state withπ chiral structure (OTC).
The director dynamics is in-plane rotational in the bulk region and out-of
plane oscillatory in the boundary layers, the directors at the upper and lower
bounding surfaces have opposite directions, and the system never recovers to
the spatial homogeneous director configuration. Figure 15 is a schematic of the
rheological phase diagram given in terms of R and Er, clearly indicating the
parametric regions where the four planar modes and the five nonplanar modes
are predicted. Finally, observation probabilities of all flow modes for all the
flow regimes are shown in Fig. 15b where the probabilities are plotted for each
flow mode, and thus for any Ericksen number, the sum of the probabilities is
1. For example, for region I, the probability of IE mode is 1 and others are
zero, and for region 4 the probability of OEA and OTP flow modes are almost
0.5, and the others are zero. Summarizing this subsection, the results presented
here predict extensive multi-stabile phenomena, involving planar, chiral achiral, steady, and time periodic nodes. The range and richness of the multistability is due to the presence of the two compatibilization mechanisms
predicted by the complete theory (Rey and Tsuji 1998; Rey and HerreraValencia 2012).
Banded Textures After Cessation of Shear
Banded texture predictions during flow and after cessation of flow have been
compared with experimental data using LE and LdG models. The consistency
between the two models’ predictions have been established (Rey and Tsuji 1998).
In the LE model of banded textures during flow, the pattern formation is driven by
the OP mode that nucleates a periodic array of elliptical splay-twist-bend inversion
wall in the velocity/velocity gradient plane with a wave-length close to the shear cell
thickness. In the LdG model of banded textures, the nucleation and growth of OP
modes give rise to a heterogeneous nonplanar orientation field that relaxes through
the formation of a periodic texture.
Figure 16 shows the out-of-plane component profile after cessation of flow, for
R = 0 at the following dimensionless times: (a) t
Ã
= 2, (b) 4, (c) 6, (d) 8, (e) 12,
(f) 16, and (g) 20. The small source of the out-of-plane component near the surface is
connected across the bulk region, and the banded texture is formed with almost the
same director configuration as that during flow. Then, the texture relaxes through the
shrinking of the director out-of-plane region.
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A. D. Rey et al.
(OTP): In this nonplanar mode, the bulk director dynamics is planar and
rotational, and in the two boundary layers, it is nonplanar and rotational, and
in the two boundary layers, it is nonplanar oscillatory. The spatial profiles of
the periodic director motion are anti-symmetric. The director field exhibits
periodic chirality, that is, after a cycle of 2π rotation of the bulk director, the
system periodically recovers the spatially homogeneous director configurations (i.e., n ffi (1, 0, 0)) for 0 y
Ã
1
8. Out-of plane tumbling-wagging composite state withπ chiral structure (OTC).
The director dynamics is in-plane rotational in the bulk region and out-of
plane oscillatory in the boundary layers, the directors at the upper and lower
bounding surfaces have opposite directions, and the system never recovers to
the spatial homogeneous director configuration. Figure 15 is a schematic of the
rheological phase diagram given in terms of R and Er, clearly indicating the
parametric regions where the four planar modes and the five nonplanar modes
are predicted. Finally, observation probabilities of all flow modes for all the
flow regimes are shown in Fig. 15b where the probabilities are plotted for each
flow mode, and thus for any Ericksen number, the sum of the probabilities is
1. For example, for region I, the probability of IE mode is 1 and others are
zero, and for region 4 the probability of OEA and OTP flow modes are almost
0.5, and the others are zero. Summarizing this subsection, the results presented
here predict extensive multi-stabile phenomena, involving planar, chiral achiral, steady, and time periodic nodes. The range and richness of the multistability is due to the presence of the two compatibilization mechanisms
predicted by the complete theory (Rey and Tsuji 1998; Rey and HerreraValencia 2012).
Banded Textures After Cessation of Shear
Banded texture predictions during flow and after cessation of flow have been
compared with experimental data using LE and LdG models. The consistency
between the two models’ predictions have been established (Rey and Tsuji 1998).
In the LE model of banded textures during flow, the pattern formation is driven by
the OP mode that nucleates a periodic array of elliptical splay-twist-bend inversion
wall in the velocity/velocity gradient plane with a wave-length close to the shear cell
thickness. In the LdG model of banded textures, the nucleation and growth of OP
modes give rise to a heterogeneous nonplanar orientation field that relaxes through
the formation of a periodic texture.
Figure 16 shows the out-of-plane component profile after cessation of flow, for
R = 0 at the following dimensionless times: (a) t
Ã
= 2, (b) 4, (c) 6, (d) 8, (e) 12,
(f) 16, and (g) 20. The small source of the out-of-plane component near the surface is
connected across the bulk region, and the banded texture is formed with almost the
same director configuration as that during flow. Then, the texture relaxes through the
shrinking of the director out-of-plane region.
306
A. D. Rey et al.