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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 .