Fréedericksz transition

Derivation

Twist geometry

:\mathbf{\hat{n}}=n_x\mathbf{\hat{x}}+n_y\mathbf{\hat{y}} :n_x=\cos{\theta(z)} :n_y=\sin{\theta(z)} Under this arrangement the distortion free energy density becomes: :\mathcal{F}_{d}=\frac{1}{2}K_2\left(\frac{d\theta}{dz}\right)^2 The total energy per unit volume stored in the distortion and the electric field is given by: :U=\frac{1}{2}K_2\left(\frac{d\theta}{dz}\right)^2-\frac{1}{2}\epsilon_0\Delta\chi_eE^2\sin^2{\theta} The free energy per unit area is then: :F_A=\int_0^d\frac{1}{2}K_2\left(\frac{d\theta}{dz}\right)^2-\frac{1}{2}\epsilon_0\Delta\chi_eE^2\sin^2{\theta},dz , Minimizing this using calculus of variations gives: :\left(\frac{\partial U}{\partial \theta}\right)-\frac{d}{dz}\left(\frac{\partial U}{\partial\left(\frac{d\theta}{dz}\right)}\right)=0 :K_2\left(\frac{d^2\theta}{dz^2}\right)+\epsilon_0\Delta\chi_eE^2\sin{\theta}\cos{\theta}=0 Rewriting this in terms of \zeta=\frac{z}{d} and \xi_d=d^{-1}\sqrt{\frac{K_2}{\epsilon_0\Delta\chi_eE^2}} where d is the separation distance between the two plates results in the equation simplifying to: :\xi_d^2\left(\frac{d^2\theta}{d\zeta^2}\right)+\sin{\theta}\cos{\theta}=0 By multiplying both sides of the differential equation by \frac{d\theta}{d\zeta} this equation can be simplified further as follows: :\frac{d\theta}{d\zeta}\xi_d^2\left(\frac{d^2\theta}{d\zeta^2}\right)+\frac{d\theta}{d\zeta}\sin{\theta}\cos{\theta}=\frac{1}{2}\xi_d^2\frac{d}{d\zeta}\left(\left(\frac{d\theta}{d\zeta}\right)^2\right)+\frac{1}{2}\frac{d}{d\zeta}\left ( \sin^2{\theta}\right)=0 :\int\frac{1}{2}\xi_d^2\frac{d}{d\zeta}\left(\left(\frac{d\theta}{d\zeta}\right)^2\right)+\frac{1}{2}\frac{d}{d\zeta}\left ( \sin^2{\theta}\right),d\zeta ,=0 :\frac{d\theta}{d\zeta}=\frac{1}{\xi_d}\sqrt{\sin^2{\theta_m}-\sin^2{\theta}} The value \theta_m is the value of \theta when \zeta=1/2. Substituting k=\sin{\theta_m} and t=\frac{\sin{\theta}}{\sin{\theta_m}} into the equation above and integrating with respect to t from 0 to 1 gives: :\int_0^1\frac{1}{\sqrt{(1-t^2)(1-k^2t^2)}},dt ,\equiv K(k)=\frac{1}{2\xi_d} The value K(k) is the complete elliptic integral of the first kind. By noting that K(0)=\frac{\pi}{2} one finally obtains the threshold electric field E_t. :E_t=\frac{\pi}{d}\sqrt{\frac{K_2}{\epsilon_0\Delta\chi_e}} As a result, by measuring the threshold electric field one can effectively measure the twist Frank constant so long as the anisotropy in the electric susceptibility and plate separation is known.

Notes

References

References

1. {{harvnb. Fréedericksz. Repiewa. 1927
2. {{harvnb. Priestley. Wojtowicz. Sheng. 1975