hence, S(θ al ) increases (or decreases) monotonically from zero (π/4) to saturate at a
β-dependent plateau S 1 (0). Figure 14b shows that flow-birefringence increases with
increasing β (or rheological shape anisotropy); this prediction can be used to estimate
β and then λ, the reactive parameter through Eq. 47.
Patterns and Textures
Banded Patterns
Banded patterns normal to the flow direction after cessation of flow form for shear
rates up to 50 s
À1 ; the pattern formation time is proportional to the square of the
applied shear, and the pattern coarsening rate at sufficiently low pre-shear is well
described by a diffusive process.
The pattern is associated with relaxation of elastic energy. For shear rates greater
than 50 s
À1 , the material flow-aligns. These features were found to be similar to
those found in non-aligning lyotropic nematic polymers (Roux et al. 1995). Using
given flow kinematics and a 1D spatial description, a full rheological phase diagram
is obtained based on the LdG for non-aligning nematics in terms of the length scale
ratio R and the Ericksen number Er. The left schematic on Fig. 15 shows the
rheological phase diagram in terms of the eight director modes across the cell
thickness. For low Er, elasticity prevails, and tumbling is arrested. The dotted
parabola corresponding to small Er, containing the out-of-plane modes, regions
3–5 display multistability. The main features of these flow modes are summarized
in the following paragraph (Fig. 15 left):
1. In-Plane elastic-driven steady state (EE): The steady state of this planar mode
arises due to the long rate order elasticity stored in the deformed tensor order
parameter field. In this planar mode, there is no orientation boundary layer
behavior because there is no flow-alignment in the bulk region
2. In-plane tumbling wagging composite state (IT): In this time-dependent planar
mode, the director dynamic in the bulk region is rotational and in the boundary
layer it is oscillatory. The boundary between the bulk region and each boundary
layer is characterized by the periodic appearance of the abnormal nematic state,
which is characterized by two equal eigenvalues of the tensor order parameter
(i.e., μ k = μ r > μ 1 ) and follows a smoothly defect-free transition from the rotation
bulk region to the fixed director anchoring at the surfaces by a director resetting
mechanism. The insert in Fig. 15 is discussed in full detail below and shows
schematics of the existing stable solutions of Eq. 2
3. In-plane wagging state (IW): In this plane mode, the director dynamics over the
entire flow geometry is periodic oscillatory with an amplitude decreasing from a
maximum at the centerline to zero at the two bounding surfaces
304
A. D. Rey et al.
β-dependent plateau S 1 (0). Figure 14b shows that flow-birefringence increases with
increasing β (or rheological shape anisotropy); this prediction can be used to estimate
β and then λ, the reactive parameter through Eq. 47.
Patterns and Textures
Banded Patterns
Banded patterns normal to the flow direction after cessation of flow form for shear
rates up to 50 s
À1 ; the pattern formation time is proportional to the square of the
applied shear, and the pattern coarsening rate at sufficiently low pre-shear is well
described by a diffusive process.
The pattern is associated with relaxation of elastic energy. For shear rates greater
than 50 s
À1 , the material flow-aligns. These features were found to be similar to
those found in non-aligning lyotropic nematic polymers (Roux et al. 1995). Using
given flow kinematics and a 1D spatial description, a full rheological phase diagram
is obtained based on the LdG for non-aligning nematics in terms of the length scale
ratio R and the Ericksen number Er. The left schematic on Fig. 15 shows the
rheological phase diagram in terms of the eight director modes across the cell
thickness. For low Er, elasticity prevails, and tumbling is arrested. The dotted
parabola corresponding to small Er, containing the out-of-plane modes, regions
3–5 display multistability. The main features of these flow modes are summarized
in the following paragraph (Fig. 15 left):
1. In-Plane elastic-driven steady state (EE): The steady state of this planar mode
arises due to the long rate order elasticity stored in the deformed tensor order
parameter field. In this planar mode, there is no orientation boundary layer
behavior because there is no flow-alignment in the bulk region
2. In-plane tumbling wagging composite state (IT): In this time-dependent planar
mode, the director dynamic in the bulk region is rotational and in the boundary
layer it is oscillatory. The boundary between the bulk region and each boundary
layer is characterized by the periodic appearance of the abnormal nematic state,
which is characterized by two equal eigenvalues of the tensor order parameter
(i.e., μ k = μ r > μ 1 ) and follows a smoothly defect-free transition from the rotation
bulk region to the fixed director anchoring at the surfaces by a director resetting
mechanism. The insert in Fig. 15 is discussed in full detail below and shows
schematics of the existing stable solutions of Eq. 2
3. In-plane wagging state (IW): In this plane mode, the director dynamics over the
entire flow geometry is periodic oscillatory with an amplitude decreasing from a
maximum at the centerline to zero at the two bounding surfaces
304
A. D. Rey et al.