Soft Matter at the Thermal Scale, Ch Frank elasticity, defects and smectic layers
Problem
A planar nematic cell of thickness d=8μm and splay constant K1=6.4pN sits exactly at its Freedericksz threshold, where the director begins to tilt with a distortion wavevector q=π/d.
((a)) At threshold the tilt amplitude is still vanishingly small, so estimate instead the elastic stiffness of this mode, the Frank energy density per radian squared of tilt, f≈21K1q2.
((b)) A rubber network (recall the entropic spring result G=nskBT) has ns=1.0×1025m−3 effective strands per unit volume at T=298K. Find its shear modulus G.
((c)) Give the ratio of the Frank stiffness from part (a) to the rubber modulus from part (b), and say in one sentence what this ratio tells you about how easily a field can distort a nematic compared with how easily a rubber network can be sheared.