Derivation

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Twist Geometry

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If a liquid crystal that is confined between two parallel plates that induce a planar anchoring is placed in a sufficiently high constant electric field then the director will be distorted. If under zero field the director aligns along the x-axis then upon application of the an electric field along the y-axis the director will be given by:

 
 
 

Under this arrangement the distortion free energy density becomes:

 

The total energy per unit volume stored in the distortion and the electric field is given by:

 

The free energy per unit area is then:

 

Minimizing this using the calculus of variations gives:

 
 

Rewriting this in terms of   and   where   is the seperation distance between the two plates results in the equation simplifing to:

 

This equation simplifies further to:

 

The value   is the value of   when  . Substituting   and   into the equation above and integrating with respect to   from 0 to 1 gives:

 

The value K(m) is the complete elliptic integral of the first kind. By noting that   one finally obtains the threshold electric field  .