Soft Matter at the Thermal Scale, Ch Frank elasticity, defects and smectic layers
Problem
A planar-aligned nematic cell of thickness d has strong anchoring that holds the director parallel to the plates, along x^, at z=0 and z=d. A voltage V is applied across the cell, producing a field E=V/d along z^ (normal to the plates) as long as the director stays undistorted. The nematic has splay constant K1 and positive dielectric anisotropy Δε=ε∥−ε⊥>0, so the field favours tilting the director towards z^. Let θ(z) be the tilt angle of the director away from x^, in the xz-plane, with θ(0)=θ(d)=0. For small θ the Frank energy density reduces to 21K1(dθ/dz)2 and the dielectric coupling energy density is −21ε0ΔεE2sin2θ, so the free energy per unit area of the cell is F=∫0d[21K1(dzdθ)2−21ε0ΔεE2sin2θ]dz.
((a)) Using the calculus of variations, derive the Euler-Lagrange equation for θ(z) and show that θ(z)≡0 is always a stationary solution for any field.
((b)) Linearise the equation about θ=0, apply the boundary conditions, and find the smallest field Ec at which a non-zero θ(z) first becomes possible. Hence give the threshold voltage Vc=Ecd in terms of K1, ε0 and Δε.
((c)) Evaluate Vc for a nematic with K1=11.0pN and Δε=8.5, and confirm explicitly that your expression for Vc does not depend on d.