A nematic fills the slab with

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Vector calculus and field geometry

Soft Matter at the Thermal Scale, Ch Frank elasticity, defects and smectic layers

Problem

A nematic fills the slab 0≤z≤d0 \le z \le d with d=5.0 μmd = 5.0\,\mu\mathrm{m}. Anchoring forces the director to rotate uniformly from vertical at the lower plate to horizontal at the upper plate, so that n(z)=(sin⁡θ(z), 0, cos⁡θ(z)),θ(z)=π2zd.\mathbf{n}(z) = \big(\sin\theta(z),\,0,\,\cos\theta(z)\big), \qquad \theta(z) = \frac{\pi}{2}\frac{z}{d}. Take the splay and bend constants as K1=6.0 pNK_1 = 6.0\,\mathrm{pN} and K3=9.0 pNK_3 = 9.0\,\mathrm{pN}. ((a)) Show that this field has no twist (n⋅∇×n=0\mathbf{n}\cdot\nabla\times\mathbf{n} = 0) and find ∇⋅n\nabla\cdot\mathbf{n} and n×(∇×n)\mathbf{n}\times(\nabla\times\mathbf{n}) in terms of θ(z)\theta(z) and θ′(z)\theta'(z). ((b)) Calculate the Frank free energy density at the midplane z=d/2z = d/2.

Browse the problem library