5CB, the nematic used throughout this discussion of Frank elasticity, clears to the…

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

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

Problem

5CB, the nematic used throughout this discussion of Frank elasticity, clears to the isotropic phase at

T_{\rm NI} = 308\,\mathrm{K}

(its clearing temperature, as discussed for the isotropic-nematic transition). Take the molecular length as

a = 1.0\,\mathrm{nm}

. ((a)) Estimate the Frank constant

K

from

K \sim k_BT_{\rm NI}/a

and compare with the measured splay constant of 5CB,

K_1 = 6.4\,\mathrm{pN}

. ((b)) A planar 5CB cell of thickness

d = 15\,\mu\mathrm{m}

has dielectric anisotropy

\Delta\varepsilon = 11.5

and strong anchoring at both walls. Using the measured

K_1 = 6.4\,\mathrm{pN}

, find the Freedericksz threshold voltage

V_c

. ((c)) At threshold the director first tilts with a half-sine profile of wavevector

\pi/d

. The tilt amplitude is still vanishingly small there, so estimate the Frank (splay) energy density per radian squared of tilt,

f \approx \tfrac12K_1(\pi/d)^2

, and compare it with the shear modulus

G = nk_BT

of a lightly crosslinked rubber network with

n = 1.0\times10^{25}\,\mathrm{strands\,m^{-3}}

298\,\mathrm{K}

(the entropic-spring result

G = nk_BT

). Give the ratio

G/f

and state in words what it shows about how weak nematic elasticity is compared with rubber elasticity.