The Ericksen number and Deborah number cascades in sheared polymeric nematics

Samples of liquid crystalline poly(γ-benzyl-glutamate) solutions are sheared between glass surfaces with gaps, d = 10-500 μ, and shearing velocities, V = 0·05-10 000 μs^-1 so that the Ericksen number E ≡ Vdγ₁/K is varied over a large range, E ≈ 1-10⁷. Here γ₁ is the rotational viscosity and K₁ is the Frank splay constant, with γ₁/K₁ estimated to be approximately 1 s μ^-2 for our samples. We observe by polarizing microscopy a sequence of transitions with increasing Ericksen number analogous to that observed in small molecule tumbling nematics: namely rotation of the director out of the shearing plane and into the vorticity direction at Vd ≈ 25 μ² s^-1, and formation of roll cells at Vd ≈ 50 μ² s^-1. The roll cells become finer with increased Vd in accord with predictions of linear stability theory using the Leslie-Ericksen equations, and at Vd ≳ 500 μ² s^-1, the cells become very irregular, producing director turbulence. The turbulence becomes finer in scale as Vd increases, reaching sub-micron, and possibly molecular scales when Vd ≧ 10⁵ μ² s^-1. At the highest velocities, transitions in orientation and texture are controlled by the Deborah number De≡λV/d, where λ is the molecular relaxation time, and uniform texture-free samples are obtained when De ≳ 5.